Author Topic: The Mathematics of Pricing Games  (Read 3942 times)

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Offline LiteBulb88

  • 4/15/2019
  • TPiR Alumnus
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  • Posts: 1453
Re: The Mathematics of Pricing Games
« Reply #15 on: January 02, 2020, 12:29:18 PM »
I could be horribly wrong, but if you get both SPs right in Master Key, you actually have a 50/50 chance of winning as compared to a 40% chance.  This is because assuming you have two picks and pick a non-car key, 2/4 of the remaining keys can win you the car.

It's actually 70%. It's easier to calculate the probability of not winning the car. We're assuming you have 2 keys. The chances that the first key is not a winning key is 3/5. Assuming that's true, the second key then has a 2/4 chance of not winning the car. Multiplying those together, 3/5 * 2/4 = 3/10 = 30% chance of not winning the car, which means you have a 70% chance of winning the car. Of course, that assumes you pick everything randomly, which contestants don't generally do (they tend to prefer the keys in the middle) and the producers respond accordingly (they tend to put the master key at either position 1 or 5.)

Overall, the probability of winning Master Key, assuming all choices are done completely at random, is 37.5%. The way to calculate that is to break it down by how many SPs are won.

0 SPs won: 0% chance of winning the car
1 SP won: 40% chance of winning the car (2/5 keys will win the car)
2 SPs won: 70% chance of winning the car, as explained above.

Then we need to calculate the probability of winning 0, 1, or 2 SPs. The chance of winning 0 SPs is 1/4 (1/2 chance of getting each one wrong.) The chance of winning 2 SPs is also 1/4 (1/2 chance of getting each one right.) Thus, the chance of winning 1 SP is 1/2 (1 - 1/4 - 1/4). So overall, the probability of winning Master Key is:

P(winning 0 SPs) * P(winning the car with 0 keys) + P(winning 1 SP) * P(winning the car with 1 key) + P(winning 2 SPs) * P(winning the car with 2 keys)
= 1/4 * 0 + 1/2 * 2/5 + 1/4 * 7/10
= 15/40
= 37.5%
« Last Edit: January 02, 2020, 12:31:20 PM by LiteBulb88 »